Polytope Duality for Families of K3 Surfaces and Coupling

Abstract

We study a relation between coupling introduced by Ebeling (Kodai Math J 29:319-336, 2006) and the polytope duality among families of K3 surfaces.

Lattice duality for coupling pairs admitting polytope duality with trivial toric contribution

“…The assertion follows by case-by-case computation using formula (1) for all polytopes obtained in Mase (2021).…”

“…We present the following data in Table 1. The number(s) in the first column are given in Mase (2021). The second and fifth columns are vertices of polytopes of and obtained by Mase (2021), and the sets in the same line are polytope-dual.…”

“…The number(s) in the first column are given in Mase (2021). The second and fifth columns are vertices of polytopes of and obtained by Mase (2021), and the sets in the same line are polytope-dual. In the third and fourth columns are the Picard lattice of the family F , resp.…”

“…Since all the weighted projective spaces with weights being in Yonemura's list are Fano, we obtain subfamilies of K 3 surfaces in the space once one finds a reflexive polytope as a subpolytope of the defining polytope of the space. In the author's recent work (Mase 2021), an existence and duality of such reflexive polytopes are studied and it is concluded that almost all coupling pair extends to a polytope-duality. Once one obtains families of K 3 surfaces which already admit several dualities, one may be interested in intrinsic properties of K 3 surfaces.…”

“…We are interested in lattice-duality originally studied by Dolgachev (1996). It is concluded by Mase (2015Mase ( , 2017) that a part of transpose-dual pairs associated to strange duality of bimodal singularities extends to lattice dual, and that some subfamilies of K 3 surfaces that are double covering of the projective plane have lattice-dual property as is studied in Mase (2021). In this paper, focusing on polytope-dual pairs associated to coupling, one may pose the following problem.…”